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Unbounded selfadjoint representations of ∗ ^\ast -algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of C ∗ {C^ \ast } -algebras. Also with the aid of this generalization of Arveson’s theorem it is shown that an operator π ( A ) ¯ \overline {\pi (A)} is affiliated with the commutant π ( A ) ′ \pi (\mathcal {A})’ of a selfadjoint representation π \pi of a ∗ ^\ast -algebra A \mathcal {A} , with A = A ∗ ∈ A A = {A^ \ast } \in \mathcal {A} , if and only if π \pi preserves a certain order structure associated with A and A \mathcal {A} . This result is then applied to obtain a characterization of standard representations of commutative ∗ ^\ast -algebras in terms of an order structure.

The notation of Gramian matrix is generalized to operators between function spaces and then used to provide an operator-valued inner product between pairs of Hilbert space-valued Banach function spaces. In this context projection operators, Schwarz's inequality, and orthonormal series are discussed.

1. INTR~DLJcT~~N The connection between the convergence of the sequence {(I - aA*A)nj, where A is a bounded linear operator on a Hilbert space H, and the iterative computation of the pseudoinverse A+ of A has been investigated by various authors [2, 19, 20, 22 and 251. Also, the convergence of the sequence {T”}, where T is a bounded linear operator on a Banach space X, has received much attention over a period of years mostly in connection with the iterativp solution of linear equations [2, 8,9, 14, 15 and 181. The sequence {T”} can be replaced by (u,(T)), h w ere each a,, is an nth degree polynomial whose coefficients are nonnegative and sum to 1 [5, 6, 16 and 171. The convergence of {T”) has also been used as a hypothesis for theorems on series represents tion of Banachspace pseudoinverses [31, 19 and 201. The purpose of this paper is twofold: To study general properties of power convergent operators (bounded linear operators on X for which {Tn) con verges in a suitable operator topology), and to apply these results to the series representation of Banachspace pseudoinverses. In Section 2, we survey some properties of power convergent operators [15, 161, and present some new results and examples on spectral analysis of these operators. Section 3 gives some necessary and sufficient conditions for power convergence of certain classes of mostly Banach-space operators, Section 4 then presents conditions of the Stein type for power convergence of Hilbert-space operators. In Section 5 we give a definition of the pseudoinverse for bounded linear operators on Banach space with arbitrary range. This definition seems to be the most natural generalization of the classical Hilbert-space pseudoinverse for operators with arbitrary ranges [12, 19,20 and 251, and agrees with the definition given in [l] and [31] in the case that the ranges are closed. We then give series representations for the pseudoinverse and partial inverse; these results have numerous applications to the iterative solution of linear operator equa446

The concept of semi-normality and hyponormality are defined for unbounded operators in Hilbert Space and Commutators AB-BA, with A unbounded and seminormal and with B bounded, are studied. Four theorems are proved. Two of them are generalizations of results of C. R. Putman.

Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.

Let E E be a Banach space and let A A be a continuous function from E E into E E . Sufficient conditions are given to insure that the differential equation u ′ ( t ) = A u ( t ) u’(t) = Au(t) has a unique solution on [ 0 , ∞ ) [0,\infty ) for each initial value in E E . One consequence of this result is that if − A -A is monotonic, then − A -A is m m -monotonic and A A is the generator of a nonexpansive semigroup of operators.

Extension of pseudo-inverse operator theory to Hilbert space operators that are unbounded and have arbitrary range

Introduction. The question of ergodicity of a semigroup of bounded linear operators on a Banach space has been reduced, by Alaoglu and Birkhoff [1],' Day [2, 3], and Eberlein [4], to the study firstly of the ergodicity of the semigroup itself and secondly, of the ergodicity of each element of the Banach space with respect to this ergodic semigroup. In the case of a bounded and commutative semigroup operating on a reflexive Banach space, the ergodicity has been established [2, Corollary 6], [4, Corollary 5.1, Theorem 5.5]. For noncommutative semigroups, Alaoglu and Birkhoff have proved ergodicity by restricting the Banach space [1, Theorem 6]. A general discussion of the noncommutative case is given by Day [3 ]. The following is an attempt to prove ergodicity, in the strong topology, for a certain noncommutative case, where we use a weakened form of the commutative law. Our method uses a fixed point theorem of Kakutani [5], and a theorem about the group of cluster points of a directed subsemigroup, which we have developed elsewhere [6]. It should be observed, however, that the theorem which is proved here, though being similar to, is not a generalization of the results mentioned above. In fact it is apparently independent of them, because we postulate here a compactness in the strong topology of operators; whereas in the analogous theorems of Day and Eberlein, a kind of weak compactness is used, by considering bounded semigroups acting on reflexive spaces, a natural legacy from the study of Hilbert space. However our theoremn is a generalization of a theorem of Yosida [7, Theorem 2 ].

Problems concerning equilibrium strategies for linear games are treated in a Hilbert space setting. First a nonlinear equation is derived for the equilibrium solutions in the player’s allowed operator feedback spaces. The results obtained from the study of this equation are then used to compute the solution to a class of differential games and to establish the uniqueness of the solution.

A representation of functions as integrals of a kernel ψ ( t ; x ) \psi (t;x) , which was introduced by Studden, with respect to functions of bounded variation in [ 0 , ∞ ) [0,\infty ) is obtained whenever the functions satisfy some conditions involving the differential operators ( d / d t ) { f ( t ) / w i ( t ) } , i = 0 , 1 , 2 , … (d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots . The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in ( 0 , ∞ ) (0,\infty ) . Some approximation operators for the approximation of continuous functions in [ 0 , ∞ ) [0,\infty ) arise naturally and are introduced; some sequence-to-function summability methods are also introduced.

The purpose of this paper is to extend an interpolation theorem for sublinear operators on Lp spaces, obtained by Calderon and Zygmund, to similar operators acting on Hp spaces, p >0 [I]. We begin by making a few definitions. Let T be a mapping defined on a class of functions that are analytic in the interior of the unit circle I w| 0, if the domain of T includes Ha and there exists a constant M, independent of F in Ha, such that jjTFjjb

Unitary groups generated by differential operators have special properties that can be used to study completeness of the set of eigenvectors of the infinitesimal generator. Unitary groups also occur in differential operator theory in another manner, associated with unitary equivalence of differential operators. We discuss what happens to At= UF 'A Ut as t approaches infinity provided that At is a differential operator for all t and A has certain properties. By a differential expression T we mean a formal expression of type 0 aiDi, where D denotes id/dx and the ai are Cr* complex valued functions with a, a nonzero constant function. The domain of the ai will be (-oo, oo ) =R. Associated with an ordinary differential expression we have many possible operators in the space L2(00, ??). We shall consider only operators which contain the minimal operator To and are contained in the maximal operation TM. The definition of these operators can be found in Goldberg [1]. By a differential operator, then, we mean an operator in L2(oo, oo) which is in between To and TM for some differential expression T. Our first lemma is proved by using an argument used in Lax and Phillips [3 1. Recall that self adjoint operators generate unitary groups according to Stone's theorem. Here we will deal with selfadjoint operators which are also differential operators. These, of course, can only arise from differential expressions which are equal to their own formal adjoint. LEMMA 1. Let H be a selfadjoint differential operator. Let Ut be the unitary group which has infinitesimal generator iH, and let f be perpendicular to the eigenvectors of H. Then there is a sequence tn approaching infinity such that for any compact subinterval J of R, fJ I UtJfI 2(x)dx approaches zero. tn is independent of f and J. PROOF. Let K denote the orthogonal complement of the eigenvectors of H. Restricting H to K, we get a selfadjoint operator H, from a dense subspace of K into K. This fact may be seen by making a series of observations, each of which is easy to prove. First, if P(A) is the spectral projection of H corresponding to the Borel set Received by the editors December 11, 1970. AMS 1970 subject classifications. Primary 47E05, 34B25.

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

In this paper it is shown that if D D and E E are continuous linear operators on a Banach space X X , then the following are equivalent: (i) B B is a right factor of A A , (ii) B B majorizes A A and (iii) the range of B ∗ {B^\ast } contains the range of A ∗ {A^\ast } .

Recently there has been a great amount of research both qualitative and quantitative on the problem of approximating functions with certain growth conditions by means of linear positive operators. This paper discusses a quantitative problem for unbounded functions, i.e., the order of approximation to unbounded functions by means of linear positive operators.The theorems presented are, in several instances, extensions of known results from the case of bounded functions to the unbounded case. Applications of these theorems are given to spline functions, which are presently the subject of intensive investigation, and to the well-known Szasz-Mirakyan-Hille operators.

The purpose of this paper is to study unbounded determinable subsets of real Banach spaces. The main results obtained are the following: (i) Characterization of unbounded, convex and circled subsets of a real Banach space B that are determinable. (ii) Characterizations of unbounded, convex and circled subsets of $L( B )$, the real Banach space of all continuous linear operators on B, that are determinable in principle and determinable in practice. Disproofs of a theorem and a conjecture of Varaiya are also given.

We prove that a normal operator on a separable Hilbert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made Hilbert-Schmidt.

Using the fact that under homogeneous endpoint conditions on a finite interval the eigenvalues of the operator ${{d^2 } / {dx^2 }}$ tend to $ - \infty $, a standard construction shows that initial boundary value problems for the backward heat equation $u_x = - u_{xx} $ are not well-posed in the sense that the solutions do not depend continuously on the data. Corresponding problems for the forward heat equation are well-posed. Abstract considerations show that the pathology of heat conduction problems is essentially due to the unboundedness of ${{d^2 } / {dx^2 }}$ rather than to the distribution of its eigenvalues. Indeed, if A is a closed, unbounded operator in a Banach space X, and if the initial value problem $u'(t) = Au(t)$, $t > 0$, $u(0) = f$, is well-posed in X, then the backward initial value problem $u'(t) = - Au(t)$, $t > 0$, $u(0) = f$, is not well-posed, even with the data f restricted to the domain $D(A)$ of A.